TITLE: Faithfulness of adjoint representation of Lie algberas
QUESTION [5 upvotes]: Are there any simple or useful conditions (necessary & sufficient) under which the adjoint representation lie algebra is faithful ? One sufficient condition is semisimplicity, but perhaps this is not necessary.

REPLY [10 votes]: The adjoint representation of a Lie algebra has a kernel which is very easy to describe: It is the center of the Lie algebra (this is easy to check directly from the definition).
So the adjoint representation is faithful iff the Lie algebra has a trivial center.